A square whose side is 2m has it's corner cut away so as to form a octagon with all sides equal find the length of each side of octagon and also find the area of the octagon?
if the cutout of each side is x, the side of the octagon is x√2.
Also, 2x + x√2 = 2, so
x = 2-√2
the octagon side s = 2(√2-1)
check: 2(2-√2) + 2(√2-1) = 2
area of octagon with side s = 2s^2 (1+√2)
so, for us, that is 2(1+√2)(2√2-2)^2 = 8(√2-1)
To find the length of each side of the octagon, we can start by considering the square. Since the side length of the square is 2m, the diagonal of the square can be found using the Pythagorean theorem.
The diagonal of the square, which is also the diameter of the octagon, can be found as follows:
diagonal = √(side^2 + side^2) = √(2^2 + 2^2) = √8 = 2√2
Now we know the diameter of the octagon is 2√2.
To find the side length of the octagon, we can use the formula:
side length = (diameter) / (√2 + 1)
Plugging in the value of the diameter, we get:
side length = (2√2) / (√2 + 1)
To simplify this expression, we can rationalize the denominator:
side length = (2√2) / (√2 + 1) * (√2 - 1) / (√2 - 1)
= (2√2 (√2 - 1)) / ((√2 + 1)(√2 - 1))
= (2√2 (√2 - 1)) / (2)
= √2 (√2 - 1)
Therefore, the length of each side of the octagon is √2 (√2 - 1) meters.
To find the area of the octagon, we can use the formula:
area = (2 + 2√2) * (side length)^2
Plugging in the values:
area = (2 + 2√2) * (√2 (√2 - 1))^2
= (2 + 2√2) * 2 * (√2 (√2 - 1))^2
= 4 (2 + 2√2) * (√2 (√2 - 1))^2
= 4 (2 + 2√2) * (2 (√2 - 1))^2
= 4 (2 + 2√2) * (4(√2 - 1)^2)
= 4 (2 + 2√2) * 4 * (√2 - 1)^2
= 4 (2 + 2√2) * 4 * (√2 - 1)(√2 - 1)
= 4 (2 + 2√2) * 4 * (2 - 2√2 + 1)
= 4 (2 + 2√2) * 4 * (3 - 2√2)
= 4 * 4 * (2 + 2√2) (3 - 2√2)
= 16 * (6 - 4√2 + 6√2 - 8)
= 16 (12 - 2√2)
= 192 - 32√2
Therefore, the area of the octagon is 192 - 32√2 square meters.